Basis set construction for molecular electronic structure
theory: Natural orbital and Gauss-Slater basis for
smooth pseudopotentials
Frank R. Petruzielo, Julien Toulouse, C. J. Umrigar
To cite this version:
Frank R. Petruzielo, Julien Toulouse, C. J. Umrigar. Basis set construction for molecular
electronic structure theory: Natural orbital and Gauss-Slater basis for smooth pseudopotentials.
Journal of Chemical Physics, American Institute of Physics, 2011, 134, pp.064104. .
HAL Id: hal-00974320
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Basis set construction for molecular electronic structure theory:
Natural orbital and Gauss-Slater basis for smooth pseudopotentials
F. R. Petruzielo1,∗ Julien Toulouse2 ,† and C. J. Umrigar1‡
1
Laboratory of Atomic and Solid State Physics,
Cornell University, Ithaca, New York 14853, USA
2
Laboratoire de Chimie Théorique,
Université Pierre et Marie Curie and CNRS,
75005 Paris, France
A simple yet general method for constructing basis sets for molecular electronic structure calculations is presented. These basis sets consist of atomic natural orbitals from a multi-configurational
self-consistent field calculation supplemented with primitive functions, chosen such that the asymptotics are appropriate for the potential of the system. Primitives are optimized for the homonuclear
diatomic molecule to produce a balanced basis set. Two general features that facilitate this basis
construction are demonstrated. First, weak coupling exists between the optimal exponents of primitives with different angular momenta. Second, the optimal primitive exponents for a chosen system
depend weakly on the particular level of theory employed for optimization. The explicit case considered here is a basis set appropriate for the Burkatzki-Filippi-Dolg pseudopotentials. Since these
pseudopotentials are finite at nuclei and have a Coulomb tail, the recently proposed Gauss-Slater
functions are the appropriate primitives. Double- and triple-zeta bases are developed for elements
hydrogen through argon. These new bases offer significant gains over the corresponding BurkatzkiFilippi-Dolg bases at various levels of theory. Using a Gaussian expansion of the basis functions,
these bases can be employed in any electronic structure method. Quantum Monte Carlo provides
an added benefit: expansions are unnecessary since the integrals are evaluated numerically.
I.
INTRODUCTION
In quantum chemistry (QC) calculations, molecular
orbitals are traditionally expanded in a combination of
primitive Gaussian basis functions and linear combinations of Gaussian primitives called contracted basis functions [1]. These basis sets cannot express the correct
molecular orbital asymptotic behavior but are used in
QC calculations to permit analytic evaluation of the twoelectron integrals [2].
Analytic integral evaluation significantly limits flexibility in basis set choice but is essential for computational
efficiency in QC calculations. However, in practice, other
basis function forms can be considered since an arbitrary
function can be expanded in Gaussians. Of course, the
fidelity of this representation is limited. An expansion in
a finite number of Gaussians cannot reproduce the exponential decay of the wavefunction at large distances or
the Kato cusp conditions [3] at nuclei, but it can mimic
these features over a finite range.
Quantum Monte Carlo (QMC) calculations [4] offer
greater freedom in choice of basis functions because matrix elements are evaluated using Monte Carlo integration. Consequently, the correct short- and long-distance
asymptotics can be satisfied exactly. For systems with
a divergent nuclear potential, Slater basis functions can
exactly reproduce the correct electron-nucleus cusp and
∗
†
‡
[email protected]
[email protected]
[email protected]
long-range asymptotic behavior of the orbitals. For calculations on systems with a potential that is finite at
the nucleus and has a Coulomb tail, Gauss-Slater (GS)
primitives [5] are the appropriate choice since they introduce no cusp at the origin and reproduce the exponential
long-range asymptotic behavior of the orbitals.
Despite shortcomings, traditional QC basis sets have
yielded good results. The natural orbitals (NOs) from a
post Hartree-Fock (HF) method are a particularly successful form of contracted function [6–9]. The simplest
NO construction involves diagonalizing the one-particle
density matrix from a ground state atomic calculation
[6]. This construction is unbalanced due to obvious bias
favoring the atom. More complicated constructions involve diagonalizing the average one-particle density matrix of several systems: atomic ground and excited states,
ions, diatomic molecules, and atoms in an external electric field [7–9]. These constructions produce excellent
results, but they are complex.
A simple but general method for constructing basis sets
for molecular electronic structure calculations is proposed
and tested here. The bases are combinations of the NOs
obtained from diagonalizing the one-particle density matrix from an atomic multiconfigurational self-consistent
field (MCSCF) calculation and primitive functions appropriate for the potential in the system. The primitives are optimized for the homonuclear dimer in coupled cluster calculations with single and double excitations (CCSD), with the intention of producing a balanced
basis set. Importantly, optimal exponents for the primitive functions are shown to depend weakly on the level
of theory used in the optimization. Additionally, results
2
show that coupling is weak between primitive functions
of different angular momenta. This enables efficient determination of optimal exponents.
The utility of the above construction is demonstrated
for the elements hydrogen through argon with the nondivergent pseudopotentials of Burkatzki, Filippi, and
Dolg (BFD) [10]. Since these pseudopotentials are finite at the nuclei and have a Coulomb tail, the GS
functions are the appropriate primitives. These pseudopotentials are chosen for demonstrated accuracy in all
cases tested and because they are accompanied by a basis set. The BFD basis [10] serves as a metric for testing the new basis. The benefits of our bases extend to
all electronic structure methods tested, including CCSD,
HF, the Becke three-parameter hybrid density functional
(B3LYP) [11], and QMC.
The main area of interest for the authors is QMC. Since
QMC results depend less on basis set than traditional QC
methods [5], only double-zeta (2z) and triple-zeta (3z)
bases are presented.
This paper is organized as follows. Basis function form
and properties are demonstrated in Sec. II. Results for
calculations with the new bases are discussed in Sec. III.
Concluding remarks are provided in Sec. IV. Supplementary material is provided on EPAPS [12].
II.
BASIS SET
The number of basis functions for each angular momentum follows the correlation consistent polarized basis
set prescription of Dunning [1]. 2z and 3z bases appropriate for the BFD pseudopotentials are generated for
the elements hydrogen through argon. Since the BFD
pseudopotential removes no core for hydrogen and helium, the 2z basis for these elements consists of two S
functions and one P function, while the 3z basis consists
of three S functions, two P functions, and one D function. Since the BFD pseudopotential removes a helium
core for the first row atoms and a neon core for the second row atoms, the remaining elements lithium through
argon have the same number of basis functions. In particular, the 2z basis consists of two S functions, two P
functions, and one D function, while the 3z basis consists
of three S functions, three P functions, two D functions,
and one F function.
The bases consist of a combination of contracted and
primitive functions. Since the BFD pseudopotentials are
finite at the origin and have a Coulomb tail, the GS functions are the appropriate primitives. With the exception
of the elements in Group 1A of the periodic table (i.e. H,
Li, and Na), the basis for each element includes a single
S contraction and a single P contraction combined with
an appropriate number of GS primitives. Only two contractions are employed to reduce the computational cost
of using this basis in QC calculations. Since elements in
Group 1A of the periodic table have only one electron
for the BFD pseudopotentials, a single S orbital is the
ground state wavefunction, and this can be obtained exactly in HF. Thus, the basis for each element in Group
1A includes a single S contraction, no P contractions, and
an appropriate number of GS primitives.
A.
Contracted Functions
A contracted basis function is a linear combination of
Gaussian primitives:
s
1
X
2(2αi )n+ 2 n−1 −αi r2 m
ci
ϕnlm (r, θ, φ) =
r
e
Zl (θ, φ),
Γ(n + 12 )
i
(1)
where r, θ, φ are the standard spherical coordinates, n
is the principal quantum number, l is the azimuthal
quantum number, m is the magnetic quantum number,
Zlm (θ, φ) is a real spherical harmonic, ci is the ith expansion coefficient, and αi is the ith Gaussian exponent. In
practice, the restriction n = l + 1 applies.
The exponents of the primitive functions that form
the contracted basis functions are determined as follows.
For each angular momentum for which a contraction is
desired, an uncontracted basis consisting of nine eventempered primitive Gaussians is generated. For each
set of uncontracted Gaussians, the minimum exponent
and even-tempering coefficient are varied to minimize
the CCSD energy of the atom using a Python wrapper
around GAMESS [13].
An assumption of weak coupling between the different
angular momenta underlies the optimization procedure.
Consequently, the uncontracted basis for each angular
momentum is optimized separately. This optimization
is performed by calculating the CCSD energy on an initially coarse grid composed of different minimum exponents and even-tempering coefficients. Once regions of
low CCSD energy are identified, a finer grid is used to obtain the final minimum exponent and even-tempering coefficient. In addition to the assumption of weak coupling,
two other properties of the problem make this global optimization possible with modest computer resources; low
dimensionality of search space and efficiency of atomic
CCSD calculations.
Next, an atomic MCSCF calculation in a complete active space (CAS) with the optimized uncontracted basis
is performed in GAMESS. For these calculations, all electrons not removed by the pseudopotential are allowed to
excite. For helium, the active space consists of the orbitals from the n = 1 and n = 2 shells. For beryllium
through neon, the active space includes the orbitals from
the n = 2 and n = 3 shells. For magnesium through
argon, the active space is composed of the orbitals from
the n = 3 and n = 4 shells, with the exception of the 4D
and 4F orbitals. A subset of the natural orbitals from the
MCSCF calculations are used as the contracted functions
of our basis.
All atomic calculations are performed in D2h symmetry since GAMESS does not permit imposition of full
3
rotational symmetry. Hence, different components of the
same atomic subshell are not necessarily equivalent. Additionally, mixing may occur among orbitals of different
angular momenta. For instance, there is mixing of S
orbitals with both D3z2 −r2 and Dx2 −y2 orbitals. This
anisotropy can be removed by averaging the different
components of a particular subshell and zeroing out the
off-diagonal blocks of the one-particle density matrix [7].
A simpler approach taken in this work is found to produce results of similar quality. For each angular momentum for which a contraction is desired, the NO with
that angular momentum which has the largest occupation number is chosen. Additionally, NO elements which
do not correspond to the dominant character of the orbital are zeroed out. For instance, an NO with large coefficients on the S basis functions and small coefficients
on the D basis functions is considered to be dominated
by S character, so the D coefficients are zeroed out. Finally, the NOs are normalized. The NOs selected in this
procedure generate the contracted functions for the basis
set. The expansions of the contractions are given in the
supplementary material [12].
B.
Gauss-Slater Primitives
GS functions [5] are defined as
(ζr)2
ϕζnlm (r, θ, φ) = Nnζ rn−1 e− 1+ζr Zlm (θ, φ),
(2)
where ζ is the GS exponent and Nnζ is the normalization factor. The restriction n ≥ l + 1 is imposed for GS
functions. For r ≪ 1, the GS behaves like a Gaussian:
2
ϕζnlm (r, θ, φ) ∼
= Nnζ rn−1 e−(ζr) Zlm (θ, φ),
(3)
and for r ≫ 1, the GS behaves like a Slater:
ϕζnlm (r, θ, φ) ∼
= Nnζ rn−1 e−ζr Zlm (θ, φ).
(4)
Consequently, GS functions introduce no cusp at the origin and can reproduce correct long-range asymptotic behavior of the orbitals.
Unlike Gaussians and Slaters, normalization of GSs has
no closed form expression. Nevertheless, normalizing an
arbitrary GS is trivial with the following scaling relation
between Nnζ and Nn1 :
Nnζ = ζ n+1/2 Nn1 .
(5)
Values for Nn1 are given in the supplementary material
[12].
Since GSs are not analytically integrable, the radial
part must be expanded in Gaussians for use in QC programs that evaluate matrix elements analytically. The
expansion is
s
3
X ζ
2(2αζi )l+ 2 l −αζi r2 m
ζ
ci
Zl (θ, φ),
re
ϕnlm (r, θ, φ) =
Γ(l + 32 )
i
(6)
where cζi is the ith expansion coefficient and αζi is the
ith Gaussian exponent. Notice that the expansion permits the case for which n 6= l + 1 for the GS function.
Additionally, the following scaling relations hold for the
expansion coefficients and Gaussian exponents:
αζi = ζ 2 α1i
(7)
cζi
(8)
= c1i .
Once the Gaussian expansions are found for unit exponents, expansions of arbitrary GSs follow immediately
from the scaling relations. For QC calculations in this
paper, GSs are expanded in six Gaussians. However, if
the purpose of the initial QC calculation is to generate
crude starting orbitals for QMC calculations in which
orbital optimization is performed, it is only necessary to
expand GS primitives in a single Gaussian. In this case,
the cost of QC calculations is the same for Gaussian and
GS primitives. The expansions of GS functions with unit
exponent in both one and six Gaussians are given in the
supplementary material [12].
As mentioned above, the restriction n ≥ l + 1 is imposed for GS functions, instead of the more familiar
n = l + 1 restriction imposed for Gaussian primitives.
This motivates construction of two types of bases. In the
first, ANO-GS, the restriction n = l+1 is enforced. In the
second, ANO-GSn, for each l there can be at most a single GS primitive with a particular n. For each additional
primitive with a particular l, n must be incremented.
For example, consider lithium. The 2z ANO-GS basis
has one S contraction, one GS-1S function, two GS-2P
functions, and one GS-3D function. On the other hand,
the 2z ANO-GSn basis has one S contraction, one GS-1S
function, one GS-2P function, one GS-3P function, and
one GS-3D function.
A caveat to the above definition of the ANO-GSn basis is that GS-2S functions are not permitted since a
single GS-2S function will introduce an undesired cusp
in the wavefunction. Additionally, the 2z ANO-GS and
ANO-GSn basis sets are identical for all elements except
lithium and sodium. When the 2z ANO-GS and ANOGSn basis sets are identical, the basis sets are referred
to as a 2z ANO-GS/GSn basis. For both lithium and
sodium, the basis sets differ because these systems have
no P contractions and instead have a second P primitive for the 2z basis. This primitive is a GS-2P for the
ANO-GS basis and a GS-3P for the ANO-GSn basis. Additionally, weak coupling between functions of different
angular momentum causes the GS-1S and GS-3D functions in the ANO-GS bases for lithium and sodium to
differ from their counterparts in the ANO-GSn bases.
However, the optimal exponents differ by less than 0.01.
Optimal exponent selection for the GS primitives is discussed now. Instead of optimizing exponents for the atom
as was done to generate the contractions, optimization
of the GS exponents is performed for the homonuclear
diatomic molecule at experimental bond length [14–22].
This advantageously produces a balanced basis set.
4
Weak coupling between GS functions of different angular momenta is assumed, so the initial optimization for
each angular momentum is performed separately. This
assumption is validated in Figure 1, which contains plots
of the CCSD energy for Si2 while varying individual GS
exponents in the 2z ANO-GS/GSn basis. Both the curve
shape and exponent value which minimizes the energy
vary little with fixed exponent value, signifying weak coupling between GS functions of different angular momentum.
-7.54
GS-2P=0.8
GS-2P=1.3
GS-2P=2.8
Energy (H)
-7.55
-7.56
-7.57
-7.58
-7.59
0
0.5
1
-7.40
3.5
4
GS-1S=0.5
GS-1S=1.5
GS-1S=3.5
-7.44
Energy (H)
1.5
2
2.5
3
GS-1S Exponent
III.
-7.48
-7.52
-7.56
-7.60
0
0.5
1
1.5
2
2.5
3
GS-3D Exponent
3.5
landscape are then handled with increasingly finer grids
until energy changes are less than 0.01 mH. During this
investigation of local minima, all angular momenta are
handled simultaneously to account for any coupling effects. Results of this optimization are shown in Figure
2. Optimal exponents for ANO-GS and ANO-GSn bases
exhibit a linear trend across each row of the periodic table. For nearly degenerate minima, the exponent following the trend in the figure is chosen as optimal, resulting
in energy increase no greater than several 0.1 mH. The
optimal GS exponents are given in the supplementary
material [12].
In some cases, the optimal exponents for primitives
with the same n and l are very close. This can lead to
large equal and opposite coefficients on these basis functions when constructing molecular orbitals. Numerical
problems could result, providing further motivation for
the ANO-GSn basis, in which each pair of n and l is
unique. However, all of our tests with the ANO-GS basis
have had no numerical problems.
Finally, the optimal primitive exponents are found to
depend weakly on the electronic structure method employed in the optimization, as demonstrated in Figure 3
for Si2 with the 2z ANO-GS/GSn basis. The globally
minimizing exponents are nearly equal in different methods. This exponent transferability to different levels of
theory is extremely attractive for a basis set.
4
FIG. 1. Change in Si2 CCSD energy for 2z ANO-GS/GSn
basis shows weak coupling between GS functions of different
angular momenta. TOP: Energy versus GS-1S exponent for
three values of the GS-2P exponent with the GS-3D exponent
fixed at its optimal value. Bottom: Energy versus GS-3D
exponent for three values of the GS-1S exponent with the
GS-2P exponent fixed at its optimal value.
The optimization is performed at the CCSD level of
theory using a Python wrapper around GAMESS. For
each angular momentum, an energy landscape is defined
by a grid of primitive exponents ranging from 0.1 to 6.0
with 0.1 spacing. Thorough investigation has revealed
that exponents larger than 6.0 are not optimal for the
systems considered. Low lying minima of this energy
RESULTS
Section II demonstrates that the ANO-GS and ANOGSn bases exhibit desirable properties. However, it remains to be shown that these basis sets produce accurate results. Fortunately, the basis set accompanying the
BFD pseudopotential serves as a metric for testing ANOGS and ANO-GSn basis quality. The BFD basis for elements in Groups 1A and 2A of the periodic table has
recently been updated [23], but the number of functions
in the new basis is inconsistent with the correlation consistent polarized basis prescription [1]. Since comparison
would be difficult, their published functions are considered in this work.
Figure 4 shows the CCSD total energy gain per electron of the ANO-GS and ANO-GSn bases over the BFD
bases [10] for atoms and homonuclear dimers of hydrogen through argon. Energy gains per electron tend to increase across each row of the periodic table. Both ANOGS and ANO-GSn bases yield energy gains for most
molecules and atoms. The energy gains per electron are
generally larger for molecules than for atoms, and larger
for the ANO-GSn basis than for the ANO-GS basis. The
energy gains for the 2z bases are generally larger than for
the 3z bases, as expected, since the energy left to recover
becomes smaller as the basis size increases.
The ANO-GS and ANO-GSn bases also produce more
accurate CCSD atomization energies than the BFD basis
for the homonuclear dimers of hydrogen through argon.
5
0
3z ANO-GSn
Exponent
5
4
3
GS-1S
GS-3S
GS-2P
GS-3P
GS-3D
GS-4D
GS-4F
2
1
Exponent
5
4
3
GS-1S
GS-1S
GS-2P
GS-2P
GS-3D
GS-3D
GS-4F
2
-10
-12
-14
0.5
1
1.5 2 2.5 3
GS-1S Exponent
3.5
4
0
0
-5
-1
-2
-10
-3
-15
-4
-20
-5
-6
0
0.5
1
1.5 2 2.5 3
GS-2P Exponent
3.5
4
0
2z ANO-GS/GSn
GS-1S
GS-2P
GS-3D
3
2
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
1
FIG. 2. Optimal exponents for ANO-GS and ANO-GSn bases
exhibit a linear trend across each row of the periodic table.
The 2z ANO-GS and ANO-GSn bases are identical for all
elements except lithium and sodium. The GS-1S and GS3D exponents for these elements each differ by less than 0.01
between 2z ANO-GS and ANO-GSn bases, so 2z ANO-GS
and ANO-GSn are shown together as 2z ANO-GS/GSn. Exponents for GS functions of P angular momentum are not
included for lithium and sodium since these elements have an
extra primitive of P angular momentum.
Change in Energy (mH)
5
Exponent
-8
-25
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
1
0
-6
0
Change in Energy (mH)
3z ANO-GS
4
-4
-18
6
0
CCSD
HF
B3LYP
-2
-16
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
0
Change in Energy (mH)
6
0
-20
-10
-40
-60
-20
-80
-30
-100
-40
-120
-140
-50
0
0.5
1
1.5 2 2.5 3
GS-3D Exponent
3.5
4
FIG. 3. Change in Si2 energy for 2z ANO-GS/GSn basis
shows optimal exponents depend weakly on electronic structure method (CCSD, HF, and B3LYP). Top: GS-1S exponent
is varied with GS-2P and GS-3D exponents fixed at their optimal values. Middle: GS-2P exponent is varied with GS-1S
and GS-3D exponents fixed at their optimal values. The large
increase in energy around an exponent of 1.0 occurs since the
P primitive and P contraction become nearly linearly dependent. Bottom: GS-3D exponent is varied with GS-1S and GS2P exponents fixed at their optimal values. For Middle and
Bottom, HF and B3LYP energy scale is on the right y-axis.
This difference in energy scale occurs since higher angular
momentum functions are less important in these effectively
single-determinant theories.
14
13
12
11
10
9
8
4
3
2
1
0
-1
Atom 2z ANO-GS/GSn
Dimer 2z ANO-GS/GSn
Atom 3z ANO-GS
Atom 3z ANO-GSn
Dimer 3z ANO-GS
Dimer 3z ANO-GSn
H
He
Li
Be
B
C
N
O
F
Ne
Na
Mg
Al
Si
P
S
Cl
Ar
Energy Gain per Electron (mH)
6
FIG. 4. CCSD total energy gains per electron of ANO-GS
and ANO-GSn relative to the corresponding BFD basis [10]
for atoms and homonuclear dimers of hydrogen through argon. Energy gains per electron tend to increase across each
row of the periodic table. The 2z ANO-GS and ANO-GSn
bases are identical for all elements except lithium and sodium.
Differences between 2z ANO-GS and ANO-GSn results for
these elements is ∼ 0.01mH, so they are shown together as 2z
ANO-GS/GSn.
Figure 5 shows the fraction of experimental atomization
energy recovered in CCSD for the homonuclear dimers
which are not weakly bound. The 2z ANO-GS/ANOGSn basis recovers more atomization energy than the 2z
BFD basis for all dimers except those of Group 1A elements. Similarly, the 3z ANO-GSn basis recovers more
atomization energy than the 3z BFD basis for the same
systems, but the differences are small. The 3z ANO-GSn
is on average slightly better than the 3z ANO-GS basis,
the largest gains being for F2 and Cl2 .
For Group 1A elements, the BFD bases recover more
atomization energy in CCSD than do their ANO-GS or
ANO-GSn counterparts. This occurs due to inaccurate
BFD energies for the atoms, as can be seen in Figure
4. However, as described above, we used the published
BFD bases for these elements rather than the updated
BFD bases [23] to maintain consistency.
Finally, improvements of the ANO-GS and ANO-GSn
bases extend to other systems and methods. Figure 6
shows the fraction of experimental atomization energy
recovered for five systems in the G2 set [26] with the
BFD, ANO-GS, and ANO-GSn bases in three quantum
chemistry methods. For CCSD, the ANO-GS and ANOGSn bases outperform the BFD basis for all systems. For
sulfur dioxide the improvement due to the ANO-GS and
ANO-GSn bases is dramatic: the 2z ANO-GS/GSn result
is nearly halfway between the 2z and 3z BFD results, and
the 3z ANO-GS/GSn result is nearly halfway between the
3z and 5z BFD results. ANO-GS and ANO-GSn benefits
are more prominent in HF and B3LYP: for most systems,
the 2z ANO-GS/GSn result is closer to the 3z BFD result
than the 2z BFD result, and the 3z ANO-GS/GSn result
is closer to the 5z BFD result than the 3z BFD result.
Differences between results with the ANO-GS and ANOGSn bases are small.
Figure 7 shows the fraction of experimental atomization energy recovered using diffusion Monte Carlo (DMC)
with the BFD, ANO-GS, and ANO-GSn bases. For each
system, the DMC calculations are performed with both
a single-configuration state function (single-CSF) reference (DMC-1CSF) and full-valence complete active space
reference (DMC-FVCAS). However, for each of the constituent atoms in these molecules, the FVCAS and singleCSF references are equivalent. All DMC calculations are
performed with a 0.01 H−1 time step and trial wavefunction obtained by optimizing Jastrow, orbital, and configuration state function (CSF) parameters (where applicable) via the linear method [28–30] in variational Monte
Carlo. The DMC-1CSF and DMC-FVCAS calculations
exhibit similar trends to the HF and B3LYP calculation
for most systems: the 2z ANO-GS/GSn result is closer
to the 3z BFD result than the 2z BFD result, and the 3z
ANO-GS/GSn result is closer to the 5z BFD result than
the 3z BFD result. Again, differences between results
with the ANO-GS and ANO-GSn bases are small.
There are several important points that can be made
by comparing the DMC calculations of Figure 7 to the
CCSD calculations of Figure 6. First, the DMC results
for the atomization energies have a weaker dependence
on basis size than the CCSD results. Second, for a given
basis set, the most basic DMC calculations, DMC-1CSF,
yield superior results compared to CCSD. In addition to
yielding superior results, DMC-1CSF calculations have
better computational cost scaling than CCSD calculations. Under certain assumptions, the cost of DMC-1CSF
calculations scales as O(N 3 ) [31], while the cost of CCSD
calculations scales as O(N 6 ) [32], where N is the number of electrons. However, it is important to note that
the prefactor of the scaling is significantly smaller for the
CCSD calculations.
Finally, our results are not the first to show that DMC
calculations can produce accurate atomization energies.
In particular, DMC-1CSF calculations of the entire G2
set have been performed for both pseudopotential and
all-electron systems [33, 34] and produced excellent results. Additionally, there is good agreement between the
pseudopotential and all-electron results with a mean absolute deviation of about 2.0 kcal/mol over the entire G2
set [34]. Although these previous results are very good,
there is room for improvement, particularly for the open
shell systems. A systematic study with DMC-FVCAS
calculations is currently underway in our group, which
should produce results to (near) chemical accuracy for
all systems in the G2 set.
Fraction of Exp. Binding Energy
7
1.2
1.1
2z BFD
2z ANO-GS/GSn
3z BFD
3z ANO-GS
3z ANO-GSn
5z BFD
H2
C2
1
0.9
0.8
0.7
0.6
0.5
0.4
Li2
B2
N2
O2
F2
Na2
Al2
Si2
P2
S2
Cl2
FIG. 5. Fraction of experimental atomization energy recovered in CCSD with BFD, ANO-GS, and ANO-GSn bases for the
homonuclear dimers of hydrogen through argon which are not weakly bound. The 2z ANO-GS and ANO-GSn bases are
identical for all elements except lithium and sodium. Differences between 2z ANO-GS and ANO-GSn atomization energies for
these elements is ∼ 0.01mH, so they are shown together as 2z ANO-GS/GSn. Calculated values are corrected for zero point
energy [17, 24] to compare with experiment [14, 16, 17, 25].
IV.
CONCLUSION
A simple yet general method for constructing basis sets
for molecular electronic structure theory calculations has
been presented. These basis sets consist of a combination
of atomic natural orbitals from an MCSCF calculation
with primitive functions optimized for the corresponding
homonuclear dimer. The functional form of the primitive
functions is chosen to have the correct asymptotics for the
nuclear potential of the system.
It was shown that optimal exponents of primitives with
different angular momenta are weakly coupled. This enables efficient determination of optimal exponents. Additionally, it was demonstrated that the particular electronic structure method employed in optimization has
little effect on the optimal values of the primitive exponents.
Two sets of 2z and 3z bases, ANO-GS and ANO-GSn,
appropriate for the Burkatzki, Filippi, and Dolg nondivergent pseudopotentials were constructed for elements
hydrogen through argon. Since these pseudopotentials
do not diverge at nuclei and have a Coulomb tail, GS
functions are the appropriate primitives.
It was demonstrated that both ANO-GS and ANOGSn basis sets offer significant gains over the Burkatzki,
Filippi and Dolg basis sets for CCSD, HF, B3LYP [11],
and QMC calculations. Improvements were observed in
both total energies and atomization energies. The latter
indicates that basis sets providing a balanced description
of atoms and molecules were produced by using both the
atom and the dimer in the optimization. On average,
the ANO-GSn basis is slightly better than the ANO-GS
basis, but either is a sound choice.
In the future, these basis sets will be extended to
include the transition metals, and, bases will be constructed for all-electron calculations, for which Slater
functions are the appropriate primitives.
V.
ACKNOWLEDGMENTS
We thank Claudia Filippi for very valuable discussions.
This work was supported by the NSF (Grant Nos. DMR0908653 and CHE-1004603). Computations were performed in part at the Computation Center for Nanotechnology Innovation at Rensselaer Polytechnic Institute.
CCSD
1.00
0.90
0.80
0.70
0.60
Fraction of Exp. Binding Energy
LiF
O2
P2
S2
B3LYP
1.00
0.90
0.80
0.70
1.10
DMC-FVCAS
1.00
0.90
0.80
0.70
0.60
SO2
1.10
LiF
LiF
O2
P2
1.00
S2
SO2
0.80
0.60
0.40
0.20
0.00
LiF
O2
P2
S2
P2
S2
SO2
P2
S2
SO2
DMC-1CSF
1.00
0.90
0.80
0.70
LiF
O2
FIG. 7. Fraction of experimental atomization energy recovered in diffusion Monte Carlo (DMC) for LiF, O2 , P2 , S2 , and
SO2 with the BFD, ANO-GS, and ANO-GSn bases. DMC
calculations are performed with both a single-CSF reference
(DMC-1CSF) and full-valence complete active space reference
(DMC-FVCAS). The 2z ANO-GS and ANO-GSn bases yield
different results only for LiF. The 5z BFD* calculations do
not include the G or H functions from the 5z BFD basis.
Calculated atomization energies are corrected for zero point
energy [17, 24] to compare with experiment [14, 16, 17, 27].
The legend for this plot is identical to that of Figure 6.
2z BFD
2z ANO-GS
2z ANO-GSn
3z BFD
3z ANO-GS
3z ANO-GSn
5z BFD*
HF
O2
1.10
0.60
0.60
Fraction of Exp. Binding Energy
Fraction of Exp. Binding Energy
1.10
Fraction of Exp. Binding Energy
Fraction of Exp. Binding Energy
8
SO2
FIG. 6. Fraction of experimental atomization energy recovered in HF, B3LYP, and CCSD for LiF, O2 , P2 , S2 , and SO2
with BFD, ANO-GS, and ANO-GSn bases. The 2z ANOGS and ANO-GSn bases yield different results only for LiF.
The 5z BFD* calculations do not include the G or H functions from the 5z BFD basis. Calculated atomization energies
are corrected for zero point energy [17, 24] to compare with
experiment [14, 16, 17, 27].
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